Transmission going out???

I have a 91 Park Ave and a have a radiator leak. Chicago weather is kicking in so its getting colder outside so driving it is to a minimum. But the question that I have is will that cause the transmission not to shift down properly? The trans still shifts smoothly so I do not know if the trans is going out or what. Momemtarily, I would come to a stop and from my rpm meter, it will still be at about 1500. I will press on the gas and it will not do anything like I have the car in neutral. After a couple secs. it will drop and I can feel the engine shake. Could the radiator be the problem and heat issues are going on or something else??

After the engine warms up, will it go into gear when you move the shift lever from P/N to (D) and will it accelerate normally?

The acceleration is fine when I shift from park to drive. It shifts smooth.

Will it shift into gear when the engine is cold, like when you first start the engine?

yes

Sorry, but you do have a problen and you should take the car to a transmission shop for further diagnosis.